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OSMANIA UNIVERSITY LIBRARYCall No. ZUtf&lkC- Accession No.,Author &UHVIi. C.' OOTitle C*ut~ie*Jl frnjsbULoyrf*y.

This book should be returned on or before the datr, last marked below.


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CHEMICALCBYSTALLOGRAPHY

AN. INTRODUCTION TOOPTICAL AND X-RAY METHODS

BYC. W. BUNN

OXFORDAT THE CLARENDON PRESS


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Oxford University Press, Amen House, London E.G. 4GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTONBOMBAY CALCUTTA MADRAS CAPETOWN

Geoffrey Cumberlege, Publisher to the University

FIRST PUBLISHED 1045REPRINTED WITH CORRECTIONS 1946

Reprinted lithographically in Great Britainat the UNIVERSITY PRESS, OXFORD, 1948, 1949,1952, from corrected sheets of the second

impression


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PREFACECEYSTALLOGRAPHIC methods are used in chemistry for two main pur-poses the identification of solid substances, and the determination ofatomic configurations ; there are also other applications, most of which,as far as technique is concerned, may* be said to lie between the twomain subjects. This baok is intended to be a guide to these methods.I have tried to explain the elementary principles involved, and to giveas much practical information as will enable the reader to start usingthe methods described. I have not attempted to give a rigorous treat-ment of the physical principles: thefapproach is consistently from thechemist's point of view, and physical theory is included only in so faras it is necessary for the general comprehension of the principles andmethods described. Nor have I attempted to give an exhaustiveaccount of any subject ; the aim throughout has been to lay the founda-tions, and to give sufficient references (either to larger works or tooriginal papers) to enable the reader to follow up any subject in greaterdetail if he so desires.The treatment of certain subjects is perhaps somewhat unorthodox.Crystal morphology, for instance, is described in terms of the conceptof the unit cell (rather than in terms of the axial ratios of the earliermorphologists), and is approached by way of the phenomena of crystalgrowth. The optical properties of crystals are described solely in termsof the phenomena observed in the polarizing microscope. X-ray diffrac-tion is considered first in connexion with powder photographs; it ismoj*e usual to start with the interpretation of the diffraction effects ofsingle crystals. These methods of treatment are dictated by the formand scope of the book ; they also reflect the course of the writer's ownexperience in applying crystallographic methods to chemical problems.It is therefore hoped that they may at any rate seem natural to thoseto whom the book is addressed -students of chemistry who wish toacquire some knowledge of crystallographic methods, and researchworkers who wish to make practical use of such methods. If the bookshould come to the notice of a more philosophical reader, I can onlyhope that any qualms such a reader may feel about its avoidance offormal physical or mathematical treatment may be somewhat offsetby the interest of a novel, if rather severely practical, viewpoint.The difficulties of three-dimensional thinking have, I hope, beenlightened as much as possible by the provision of a large number of


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vi PREFACEdiagrams ; but crystallography is emphatically not a subject which canbe learnt solely from books: solid models should be used freely modelsof crystal shapes, of atomic .nd molecular configurations, of reciprocallattices and of vectorial representations of optical and other physicalproperties. Most of the diagrams are original, but a few have beenreproduced, by permission, from published books and journals: Figs.197, 207-9, 215, and 222 from the Journal of the Chemical Society ;Figs. 199, 203, and 217 from the Proceedings of the Royal Society,Figs. 102-4 from the Journal of Scientific Instruments ; Fig. 229 fromthe Journal of the American Chemical Society, Fig. 161 from Inter-nationale Tabetten ftir Bestimmung von Kristallstrukturen (Berlin: Born-traeger); Fig. 192 from the * Strnkturbericht ' of the Zeitschrift fttrKristallographie\ and Figs. 212 and 216 from Bragg's The CrystallineState (London: G. Bell & Sons, Ltd.). For Figs. 220-1 I wish to thankMessrs. G. Huse and H. M. Powell. Finally I wish to thank Dr. F. C.Phillips and his colleagues at the Department of Mineralogy andPetrology, Cambridge, for permission to use* their scheme of exhibitingthe relations between the crystal classes by miniature stereographicprojections (Fig. 32).I have great pleasure in acknowledging the help of my friends andcolleagues, and proclaiming my gratitude for it. First of all I wish tothank Professor C. N. Hinshelwood, at whose suggestion the book waswritten, and whose interest and encouragement stimulated its progress.Next I must thank Mr. H. S. Peiser, who read the whole work in manu-script, made many valuable suggestions, contributed the geometricalproofs of appendixes 2 and 4, and compiled the subject index. Parts ofthe book were read by Mr. R. Brooks, Dr. L. M. Clark, and Mr. T. C.Alcock ; their suggestions were gratefully received. I am also indebtedto Dr. H. Lipson for a discussion on nomenclature. In checking thetypescript and proofs I have been very much helped by my wife, byMr. C. A. Smale, and Miss A. Turner-Jones. The last-mentioned andMr. H. Emmett kindly drew some of the diagrams. The X-ray photo-graphs were, with one exception, taken by Mr. J. L. Matthews andMr. T. C. Alcook, and printed by Mr. W. J. Jackson ; the exception isthe Weissenberg photograph of Plate VIII, for which I am indebtedto Messrs. R. C. Evans and H. S. Peiser. The photomicrographs andoptical diffraction photographs (Plates I, II, V, and XIII) were takenby Mr, H. Emmett.

Finally, I wish to say that the experience on which the book is basedwas gained in the Research Laboratory of I.C.I. Limited (Alkali


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PBEFACE viiDivision) at Northwich. The support and encouragement of Mr. H. E.Cocksedge (formerly Research Manager), of his successor Dr. J. C.Swallow, of the present Research Manager, Dr. J. Ferguson, and ofmany of my colleagues more especially Dr. L. M. Clark and Mr. E. A.Cooke are gratefully acknowledged. C.W.B.


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CONTENTSI. INTRODUCTORY SURVEY .... 1

Anisotropy .... .2Identification of crystals under the microscope . . 3Origin of anisotropic properties of crystals . . .4Molecular type and arrangement deduced from anisotropic

properties of crystals . . , . .5The use of X-rays . . . . .6Electron density maps . . . . .6Limitations of X-ray methods . . * .7Use of X-ray diffraction patterns for identification . . 8Information obtainable by partial interpretation of X-raydiffraction patterns . , , . 9Value of using more than one method . . .9Plan of this book . . . . .9

SECTION 1. IDENTIFICATIONII. THE SHAPES OF CRYSTALS . . , .11

Shape varies with conditions of grc vvth . . .12The unit of pattern ('unit cell 5 ) . . . .14Crystal growth . . . . . .18Nomenclature of crystal planes . , . .24The law of rational indices . . . .27Measurement of interfacial angles, and graphical representation 28Deduction of possible unit cell shape from crystal shape. Pre-

liminary . . . . . .30Internal symmetry and crystal shape . . .34Nomenclature of symmetry elements and crystal classes . 44The thirty-two point-group symmetries or crystal classes . 45The unit cell types or crystal systems . . .47Deduction of a possible unit cell shape and point-group sym-metry from interfacial angles . . . .62

Identification by shape . . . . .64Twinning . . . . . .67Cleavage . . . . , .68Polymorphism . . . . . .69Isomorphism and mixed crystal formation . . .69Oriented overgrowth . . . . .60


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CONTENTS ixin. THE OPTICAL PROPERTIES OF CRYSTALS . . 62

Cubic crystals . . . . . .63Measurement of refractive index under the microscope . 63Tetragonal, hexagonal, and trigonal crystals. Preliminary . 65Use of crossed Nicols. Extinction directions. Interference

colours . . . . . .67The indicatrix . , . . . .70Orthorhombic crystals . . . . .73Monoclinic and triclinic crystals . . . .75Use of convergent light . . . . .77Use of the quartz wedge . . . . .80Dispersion . . . . . .83Pleochroism . t . . .85Rotation of the plane of polarization . . .87Optical properties of twinned crystals . . .89

IV. IDENTIFICATION OF TRANSPARENT CRYSTALS UNDERTHE MICROSCOPE . . . . .91Cubic crystals and amorphous substances . . .93

Optically uniaxial crystals . . . . .94Optically biaxial crystals . . . . .96Mixtures . . . . . .99Identification when it is not possible to measure refractive indices 100

V. IDENTIFICATION BY X-RAY POWDER PHOTOGRAPHS . 103The production of X-rays . . . .103X-ray wave-lengths . . . . .105X-ray powder photographs . . . .108Powder cameras . . . . . .109General characteristics of X-ray powder photographs , 112Diffraction of X-rays by a crystal . . . .114Measurement of powder photographs . . .119Spacing errors in powder photographs . . .120Identification of single substances, and classification of powderphotographs . . . . . .122

Identification and analysis of mixtures . . .123Non-crystalline substances . . .127


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; CONTENTSSECTION 2. STRUCTURE DETERMINATION

VI. DETERMINATION OF UNIT CELL DIMENSIONS . .128Cubic unit cells . . . . . .129Tetragonal unit cells . . . . .132Hexagonal, trigonal, and rhombohedral unit cells . .134Other types of unit cells . . . . .136Single-crystal rotation photographs . . .137Unit cell dimensions from rotation photographs . .139Indexing rotation photographs. Preliminary consideration . 142The 'reciprocal lattice' . . . . .144Indexing rotation photographs by reciprocal lattice methods*.Orthorhombic crystals . n . . . .151Monoclinic crystals . . . . .153Triclinic crystals . . . . . .156Oscillation photographs . . . . .158The tilted crystal method . . . .162Moving-film goniometers . . . . .166The simplest unit cell , . . . .172The accurate setting of ill-formed crystals . . .173Oriented polycrystalline specimens . . . .175Determination of unit cell dimensions with the highest accuracy 1 80Application of knowledge of unit cell dimensions :

1. Identification . . . . .1812. Determination of composition in mixed crystal series . 1833. Determination of molecular weight . . .1854. Shapes of molecules, and orientation in the unit cell . 1 875. Chain-type in crystals of linear polymers . 188

VII. DETERMINATION OF THE POSITIONS OF THE ATOMSIN THE UNIT CELL BY THE METHOD OF TRIAL ANDERROR . . . . . . .190Measurement of X-ray intensities . . . .192Calculation of intensities. Preliminary . . .196The diffracting powers of atoms . . . .196The structure amplitude, F . . . .196The number of equivalent reflections, p . . .199Angle factors ...... 200Thermal vibrations ..... 204Absorption ..... 206Complete expression for intensity of reflection. Perfect and

imperfect crystals. ..... 207


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IINTRODUCTORY SURVEY

MOST solid substances are crystalline, that is to say, the atoms or mole-cules of which they are composed are packed together in a regularmanner, forming a three-dimensional pattern. In some solids manyminerals, for instance the fact that they are crystalline is obvious tothe unaided eye ; the plane faces and the more or less symmetrical shapeof the particles are evidence of an orderly internal structure. In othersolids all we see is a powder or some irregular lumps ; but with the aid ofthe microscope and the still more dedicate X-ray methods we have cometo realize that most of the solids with which we are familiar, from rocksto sand and soil, from the chemical reagents on our laboratory shelvesto paint pigments and cleaning powders, from steel and concrete tobones and teeth, really consist of small crystals. Even such apparentlyunlikely materials as wood, silk, and hair are at any rate partly crystal-line ; the molecules composing them are to some extent packed togetherin an orderly way, though the regularity of arrangement is not main-tained throughout the whole of the material.The crystalline condition is, in fact, the natural condition in the solidstate ; at low temperatures atoms and molecules always try to arrangethemselves in a regular manner. When they do not succeed in doing sothere is good reason for their failure. Some glasses, for instance, aresiipercoolcd liquids in which crystals have not been able to grow owingto very rapid cooling and the very high viscosity of the liquid ; low-temperature decomposition products such as 'amorphous' carbon areformed at such temperatures that atomic movements are too sluggishto permit crystal growth; some polymers (such as 'bakelite') are com-posed of molecules which are large and irregular in structure and cannotpack together neatly.

(Even in these 'amorphous' substances it is by no means certain thatorder is entirely lacking. The word 'amorphous' has to be used withcaution and inverted commas, for some people consider that glassesand low-temperature decomposition products are really composed ofextremely small crystals only a few atoms across ; moreover, some ofthe macromolecular polymers like rubber, which are 'amorphous' in theordinary condition at room temperature, can be brought to a crystal-line condition by stretching. Even in liquids disorder is not complete ;there is SOTIIO attempt to form a regular arrangement. An interesting

4458 B


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2 INTRODUCTORY SURVEY CHAP. Iaccount of work on these substances up to 1934 is to be found in J. T.Randall's book, The Diffraction of X-rays and Electrons by AmorphousSolids, Liquids, and Oases.)The fact that in most solid substances the atoms or molecules are

arranged in an orderly manner is of great significance for the chemist,whether he is a philosopher in a university or an analyst in an industriallaboratory. The chemist is interested in such things as the structureof molecules, the nature of the bonds between dtoms, and the arrange-ment of ions ; and he uses every property of a substance which can givehim any information on these matters. He is also inevitably concernedwith methods for the identification and analysis of the substances heencounters. Crystals, in virtue of the orderly arrangement of the atomsor molecules composing them, have very special properties, which notonly make possible the most precise determinations of molecular struc-tures, but also provide powerful and certain methods of identificationand analysis.Anisotropy . To begin with, the properties ofa crystal are, in general,

not the same in all directions. A crystal grows, not as a sphere, but asa polyhedron; it dissolves morequickly in some directions than inothers ; its refractive index (exceptin certain special cases) varies withthe direction of vibration of thelight waves ; its magnetic suscepti-bility, its cohesion, its thermal ex-QO GO pansion, its electrical conductivity,

'QQ Q) aU vary with direction in thecrystal. This variation of proper-FIG. 1. Crystal properties vary with tieg with crygtal direction, Or 'ani-direetion. , . J _ _sotropy , is a consequence of the

regular packing of atoms or molecules in a crystal. In a normal liquidor a gas the atoms or molecules are oriented at random, and con-sequently the properties are the same in all directions; individualmolecules may be strongly anisotropic, but owing to the randomorientation of the large numbers of molecules present even in micro-scopic samples, the properties are averaged out in all directions. Ina crystal the atoms are drawn up in ranks ; pass through it in imagina-tion, first in one direction and then in another, and (unless you havechosen two special directions) you will encounter the constituent atomsor molecules at different intervals and perhaps (if there are different


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CHAP, i INTRODUCTORY SURVEY 3kinds of atoms) in a different order in the two directions. (See Fig. 1,a two-dimensional analogy.) Since the arrangement of the atoms ormolecules in a crystal varies witjh direction, certain properties of thecrystal must also vary with direction.

Crystals thus have a greater wealth and variety of measurable charac-teristics than liquids or gases. This circumstance can be turned to goodaccount ; we can use these varied directional properties for the identi-fication of crystalline siibstances. Since there are more characteristicmagnitudes to determine, identification by physical methods is im-mensely more certain for crystals than it is for liquids or gases.

Identification of crystals under the microscope. Of the charac-teristics which are most useful for identification purposes the mostreadily determined are shape and refractive indices. The determinativemethod which has proved most valuable for microscopic crystals (suchas those in the average experimental or industrial product) is to measurethe principal refractive indices (up to three in number, depending on thesymmetry of the crystal) and, if jjossible, to find the orientation ofthe principal optical directions with respect to the geometrical form ofthe crystal. This information, which can all be obtained by simple andrapid microscopic methods, is usually sufficient to identify any crystal-line substance whose properties have previously been recorded. Mix-tures oftwo or more crystalline substances can be identified by the samemethod ; in phase equilibrium studies and in industrial research it is notuncommon to encounter mixtures of three or four constituents, all ofwhich can be identified in this way.

This method of identification sometimes has certain advantages overchemical analysis. A single substance can often be identified in a fewminuteswhere a chemical analysis might take hours, and only very smallquantities of material are required. But in general the method must notbe regarded as a rival to chemical analysis but as a valuable complement.It gives essential information in cases where chemical analysis does nottell the whole story or does not even touch the most important part ofthe story. Where substances capable of crystallizing in two or moredifferent forms are concerned (for instance, the three forms of calciumcarbonate calcite, aragonite, and vaterite), chemical analysis cannotdistinguish between them, and a crystallographic method is essential.The greatest advantages, however, are shown in the analysis of mixturesof several solid phases. Chemical analysis tells us which atoms or ionsare present, as well as the proportion of each, but it does not usually tellus which of these are linked together. For instance, a solid obtained in a


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4 INTRODUCTORY SURVEY CHAP, iphase equilibrium study of the reciprocal salt pair NaN03-KCl-(H20)is shown by chemical analysis to contain all four ions, Na, K, N03 , 01,in certain proportions. But which substances are present ? NaCl,NaN03 ,and KNO8 , or NaCl, NaNO3 , and KC1, or perhaps all four possible salts,NaCl, NaN03 , KC1, KN03 ? This question can be most readily settledby a ocystallographic method of identification. As another example,consider a refractory material whose composition can be represented asso much alumina and so much silica; are these present as separateconstituents or are they combined as an aluminium silicate ? If theyare combined, which ofthe several known aluminium silicates is present ?And is the material all aluminium silicate, or is there some excess silicaas well as an aluminium silicate ?


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CHAP. I INTRODUCTORY SURVEY 5the molecules or polyatomic ions are packed. In some crystals all themolecules are packed parallel to each other, and these crystals haveproperties which correspond with those of individual molecules. Acrystal composed of long molecules all packed parallel as in Fig. 2 a(a crystal of a long-chain hydrocarbon, for instance) has a high refrac-tive index for light vibrating along the molecules, and low refractiveindices for light vibrating in all directions perpendicular to the mole-cules. In other crystals*the molecules are not all parallel to each other ;sometimes half the molecules have one orientation and half anotherorientation, as in Fig. 2 6 ; sometimes the arrangement is still more

00no*OUOUMB 9^9 9w

FIG. 2. a. Long molecules packed parallel, b. Long molecules arranged sothat there are two different orientations, c. In some crystals composed ofmonatomic ions, anisotropy results from the mode of packing of the ions.

complex (it depends on the shape of the molecules and the intermole-cular forces). The properties of these crystals correspond, not withthose of a single molecule, but with those of a small group of two ormore differently oriented molecules.To turn now to crystals composed of 'unattached* atoms ormonatomicions, which are individually isotropic. Here it is only the second factor

the effect of arrangement which can be responsible for anisotropyin the crystal. It is the orderliness of arrangement itself which, becauseit gives rise to different atomic distributions in different directions(Fig. 2 c), confers properties varying with crystal direction. The degreeof anisotropy is usually far less in these crystals than in crystals con-taining molecules or polyatomic ions which are themselves anisotropic.Molecular type and arrangement deduced from anisotropicproperties of crystals. It is evident that, in dealing with crystals ofunljpiown structure, the anisotropic properties may often be used togive direct information about the general shape of the molecules orpolyatomic ions in the crystals and the way in which the molecules orions are packed. A strongly anisotropic crystal must contain stronglyanisotropic molecules or polyatomic ions packed in such a way that the


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6 INTRODUCTORY SURVEY CHAP. Ianisotropies of the different molecules or ions do not neutralize eachother, and a consideration ofthe properties ofthe crystal in all directionsmay lead to a fair idea of the general shape of the molecules or ions andthe way they are packed. This use of optical and other properties togive information about molecular or ionic shape and arrangement is astriking example of the advantages conferred by the ordered structureof crystals. A molecule is too small to study individually by methodsavailable at the present time ; but a crystal, in'which a large number ofmolecules are packed in a regular manner, is in a sense a vastly enlargedmodel of a molecule or a small group of molecules, and when we observethe optical properties of such a crystal under the microscope, we areobserving in effect the optical properties of a molecule or a small groupof molecules, and this may tell us something about the shape of themolecules and the way they are packed in the crystal.The use of X-rays. All the information mentioned hitherto is ob-tained by old and well-established methods, of which by far the mostimportant and generally useful is the determination of optical propertiesunder the microscope. Visible light, however, gives us only a rough ideaof the internal structure of a crystal ; its waves, being much longer thanthe distances between atoms, are much too coarse to show the details. Ifwe want a more detailed picture of the structure of molecules and thearrangement of atoms and ions, as well as a yet more powerful methodof identification, we must use much shorter waves, of about the samelength as the distances between atoms. The X-rays, produced whenhigh-speed electrons hit atoms, happen to be about the right length.The discovery of this fact, due to Laue in 1 912, was of course one of themost important -discoveries in the present century ; it opened the way,not only to an understanding of the nature of X-rays, but also to thedetermination of the exact arrangement of the atoms in crystals. True,we cannot get a direct image of the atomic pattern in a crystal ; X-rayscannot be focused in the convenient ways used for visible light. Whatwe have to do is to study the diffraction effects produced when X-rayspass through a crystal, and build up an image of the structure by calcu-lation. The diffraction of X-rays by crystals is not essentially differentfrom that of visible light by a diffraction grating ; but to synthesize theimage from the diffracted waves we must use, not lenses, but equations.Electron density maps. Since it is the electrons in the atoms which

t Recently,W. L. Bragg ( 1 939, 1 942 a) has shown that, starting with the data providedby tho X-ray diffraction pattern, an image can be formed experimentally by a methodemploying visible light : the interference of light waves takes the place of calculations.


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CHAP, i INTRODUCTORY SURVEY 7are responsible for the diffraction of X-rays, the image we build up bycalculation is a sort of contour map of electron densities in the crystal.Two or three such maps or projections, giving views of the structurefrom two or three different directions, are sufficient to enable us tobuild a complete space model ofthe crystal structure, showing the exactposition ofevery atom. The different sorts ofatoms can be identified bytheir different electron densities. The value ofsuch a model is obviouslyenormous. The exact arrangement ofions and their distances apart (giv-ing the coordination numbers and sizes' of the ions) ; the exact spatialconfiguration and interatomic distances in polyatomic ions andorganic molecules (with all that this tells us about the specific propertiesof these bodies and the nature of \be bonds between the atoms) ; themode of packing of molecules (which depends on the shape and theintermolecular forces) these are some of the fundamentals revealedat once by such a model. In the words of Bernal and Crowfoot (1933 c),the intensive analysis of X-ray diffraction patterns 'is one of the chiefmeans of transformation from the classical qualitative, topologicalchemistry of the nineteenth century to the quantum-mechanical,metrical chemistry of the present day'.Limitations of X-ray methods. If it were possible to find thestructure of every crystalline substance in this way, chemists would nolonger have to spend their time in deducing the structures of newsubstances by more or less indirect methods ; they could turn all theirenergies to preparation and synthesis. In the future it may well happenthat the structures of crystals will be determined by X-ray methodswithout chemical evidence of any sort, but at the present time there arecertain difficulties which restrict the scope of such methods.As may be imagined, the building by calculation of an image of thepattern of atoms in a crystal is a complex and lengthy task. Moreover,it is not (except in special cases) straightforward ; that is to say, wecannot proceed straight from the experimental data (the positions andintensities ofthe diffractedX-ray beams) to the calculation of the image ;at one stage it is nearly always necessary to use the procedure of trialand error, that is, to think of an atomic arrangement, calculate thediffraction effects it would give, and compare these with the actualdiffraction effects observed ; if they do not agree, another arrangementmust be tried, and so on. Only when the approximate atomic positionshave been found in this way is it possible to calculate the final image inall its details from the experimental data. For the simpler structuresthis does not present any great difficulties, but for the more complex


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8 INTRODUCTORY SURVEY CHAP. Istructures much depends on the extent of knowledge available at thetime for the building of trial structures. In the early days of X-raycrystallography the structures of only elements and simple salts couldbe tackled with any hope of success, but with the accumulation ofknowledge, structures of ever-increasing complexity have been success-fully worked out. Up to the present time (1944) many inorganic struc-tures of considerable complexity (such as the silicate minerals, the alums,and the hetero-polyacids like phosphotungstic acid) have been workedout completely. Among organic compounds progress was at first slower,but as soon as the structures of the principal fundamental types ofmolecules (normal paraffin chain, benzene ring, naphthalene nucleus)were well established, the pace accelerated, and recently, the structuresof such complex substances as dyestuffs, carbohydrates, sterols, andhigh polymers have been solved, and even substances of extreme com-plexity (proteins) are being actively studied by this method. X-rayanalysis at first merely confirmed the conclusions of organic chemistry,but now it plays a useful part in research on chemical constitution.Use of X-ray diffraction patterns for identification. Even whencomplete structure determination is not possible, however, muchvaluable information of a less detailed character may be obtained byX-ray methods. In the first place, the diffracted beams produced whenX-rays pass through crystals may be recorded on photographic films orplates, and the patterns thus formed may be used quite empirically,without any attempt at interpretation, to identify crystalline sub-stances, in much the same way as we use optical emission spectra toidentify elements. Each crystalline substance gives its own character-istic pattern, which is different from the patterns of all other substances ;and the pattern is of such complexity (that is, it presents so manymeasurable quantities) that in most cases it constitutes by far the mostcertain physical criterion for identification. The X-ray method ofidentification is of greatest value in cases where microscopic methodsare iradequate; for instance, when the crystals are opaque or are toosmall to be seen as individuals under the microscope. The X-raydiffraction patterns of different substances generally differ so muchfrom each other that visual comparison without precise measurementis usually sufficient for identification ; but in doubtful cases measure-ment of the positions of the recorded diffractions may be necessary.Mixtures of two or more different substances which are present asseparate crystals give X-ray diffraction patterns consisting ofthe super-imposed patterns of the constituents.


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CHAP, i INTRODUCTORY SURVEY 9Information obtainable by partial interpretation of X-ray

diffraction patterns. Between the recording of an X-ray diffractionpattern and the elucidation of the complete atomic arrangement thereare several well-defined stages. Arrival at each stage gives more andmore intimate information about the substance in question. It may bepossible to form conclusions about the degree of purity of a substance,to determine its molecular weight more accurately than by any othermethod, to discover something about the symmetry of the molecules orions in the crystal, or to determine the overall dimensions of the mole-cules. Individual circumstances, the nature of the substance, and thesize and form of the crystals determine in each case how far it is possibleor desirable to go.Value of using more than one method. It must be emphasizedthat the combination of different lines ofevidence is often ofmuch greatervalue than any single method ofapproach. X-ray methods should neverbe used alone ; the combination of evidence given by X-ray diffractionpatterns with that given by optical properties, habit, cleavage, and soon may lead to valuable conclusions in circumstances where each ofthese lines of evidence taken by itself would leave unresolved ambi-guities.Plan of this book. It will be evident from the foregoing survey ofthe principal applications of crystallographic methods to chemical prob-lems that these applications fall into two classes: firstly, the use ofcrystal properties lor the purpose of identifying substances ; secondly,the elucidation of the internal structure of crystals by interpretationof their properties. This natural division determines the plan of thisbook, which is in two main sections, on identification and internalstructure respectively.

Section I (on identification) comprises four chapters. Chapter IIis an introduction to the shapes of crystals and the relation betweenshape and structure, and Chapter III is an elementary account ofcrystal optics ; some knowledge of both subjects is essential, not onlyfor the identification of crystals by microscopic methods, but also forthe understanding of the problems of structure determination dealtwith in Section II. Chapter IV deals with procedure in microscopicmethods of identification.

Chapter V, on identification by X-ray methods, is concerned withthe practical details of taking X-ray powder photographs, and alsoincludes elementary diffraction theory, taken as far as is necessary formost identification problems.


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10 INTRODUCTORY SURVEY CHAP.Section II deals, in six chapters, with the principles underlying the

progressive stages in the elucidation of internal structure. Chapters VIand VII deal with the principles of structure determination by trial ;Chapter VIII with the use of physical properties (such as habit, cleavage,and optical, magnetic, pyro- and piezo-electric properties) as auxiliaryevidence in structure determination. In Chapter IX are to be foundseveral examples of the derivation of complete structures. Chapter Xgives an introductory account of the use c*f direct and semi-directFourier series methods of building electron density maps and vectordiagrams from X-ray diffraction data.

Certain crystals give diffuse X-ray reflections ; there are variouspossible causes for this small costal size, structural irregularities, orthermal movements. The consideration of these phenomena in ChapterXI leads on to a brief introduction to the interpretation of the verydiffuse diffraction patterns given by non-crystalline substances.


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SECTION I. IDENTIFICATIONII

THE SHAPES OF CRYSTALSANYONE who has seen the well-formed crystals of minerals in ourmuseums must have boen impressed by the great variety of shapesexhibited : cubes and octahedra, prisms of various kinds, pyramids anddouble pyramids, flat plates of various shapes, rhornbohedra and otherless symmetrical parallelepipeda, and many other shapes less easy todescribe in a word or two. These crystal shapes are extremely fascinat-ing in themselves ; artists (notably Diirer) have used crystal shapes forformal or symbolic purposes, while many a natural philosopher has beendrawn to the attempt to understand first of all the geometry of crystalshapes considered simply as solid figures, and then the manner in whichthese shapes are formed by the anisotropie growth of atomic and mole-cular space-patterns.But this book has a practical object, as its title proclaims. Our pur-

pose in this chapter is to inquire to what extent crystal shapes can becriteria for identification, and how much they tell us about the atomicand molecular space-patterns within them.

In view of the great variety of crystal shapes and the rich face-development on many crystals, it is natural to expect that, on the basisof accurate methods of measurement and a sound system of classifi-cation, it would be possible to identify crystals by their shapes alone ;and indeed, in recent years attempts have been made, first by Fedorovand later by Barker and his school, to develop such a method resting onthe measurement of the angles between face-normals. There is no doubtthat when well-formed crystals, large enough to be handled individuallyso that they can be mounted on a goniometer, are available, this morpho-logical method of identification is a practicable one; Barker (1930) hasdemonstrated this. But as a standard method of identification in achemical laboratory it has very serious limitations. One of them isthat the crystals formed in laboratory experiments or in industrialprocesses are often too small to be handled individually ; they can onlybe examined under the microscope, and under these conditions angularmeasurements either cannot be made at all, or if they can be made areonly approximate. Another is that the shapes of such crystals are oftennot sufficiently characteristic; sometimes there are too few faces on


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12 IDENTIFICATION CHAP, neach crystal ; or perhaps the substance grows in the form of skeletalcrystals without definite faces; or, worse still, the crystals may bebroken into irregular pieces. To identify such materials we need amethod which does not depend on shape, but on some characteristics ofthe crystal material itself properties of the atomic space-pattern.The properties most conveniently measured under the microscope arethe optical constants, particularly the refractive indices ; and in practicethe measurement of refractive indices has provfed by far the most usefulsingle method of identifying crystalline substances under the micro-scope. The technique is described in the next chapter.

There is no need, however, to ignore crystal shape in identifica-tion work. On the contrary, whenever crystals do show good face-development their shapes, even if they cannot be measured preciselybut only observed in a qualitative way, reinforce and implement theevidence provided by optical properties, especially if the relationsbetween the principal optical and geometrical directions can bediscovered.

This is one reason for studying crystal shapes. Another and moreweighty reason is that crystal shapes tell us a great deal about therelative, dimensions and the symmetries of the atomic and molecularspace-patterns constituting the crystalline material.

In this chapter, therefore, we make some inquiry into the origins ofcrystal shapes and their classification on the basis of symmetry charac-teristics.Shape varies with conditions of growth. The shape of a crystal,

taken as it stands, is not a fixed characteristic of the substance inquestion. In the first place, the shape is controlled to some extent bythe supply of material round the crystal during growth. In uniformsurroundings, as in a stirred solution, crystals of sodium chloride growas cubes, but if they grow, well separated, on the bottom of a dish ofstagnant solution, theygrow as square tabletswhose thickness is not morethan half their other dimensions ; the reason is that growth can occuronly upwards and sideways, not downwards. If the crystals on thebottom of the dish are crowded, the tablets formed are not all square ;many have unequal edges owing to local variations in the supply ofsolute. As another example, sodium chlorate, NaClO3 , when grownrapidly in a stirred solution, forms cubes, but when grown very slowly ina still solution grows in the form of a modified cube showing additionalfacets on the edges and corners (Fig. 3). Crystals which grow inrod-like forms such as gypsum, CaS04.2H20, which is also illustrated


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CHAP. II SHAPES OF CRYSTALS 13

(b)

in Fig. 3 usually tend to grow longer and thinner when formedrapidly than when growth is slow.These are, comparatively speaking, minor variations of shape ; but

the crystal shapes of some substances may be completely altered by thepresence of certain other substances in the mother liquor. Sodiumchloride grows from a pure solution in the form of cubes, but if themother liquor contains 10 percent, of urea, the crystals whichgrow (Fig. 3) are octahedra(Gille and Spangonberg, 1927).Yet the internal structure thepattern of atoms of this sub-stance is. not changed by suchdiffering external conditions ; itis only the form of the boundingsurface ofthe crystalline materialwhich is changed. It is evidentthat if we want to use crystalshapes for identificationwe must,so to speak, get behind the shapeas it stands, and try to deducefrom the actual shape somethingabout the internal structure.

FIG. 3. Variation of crystal shape with con-ditions of growth. Sodium chlorate, NaClO3 ,grown (a) rapidly and (b) slowly; gypsum,CaSO4 . 2H2O, grown (c) slowlyand (d) rapidly;sodium chloride, NaCl, grown (c) from puresolution and (/) from sohition containing 10

per cent, of urea.

The possibility of doing this isindicated by the fact that theangles between the faces of thelong thin gypsum crystals inthe sketch are exactly the sameas those of the shorter crystals.Likewise, all octahedra of sodium chloride, however much they differin size, and however unequal the areas of the different faces of anyone crystal may be, have exactly the same interfacial angles. The slopesof the various faces are in fact controlled by the rigid, precise internalstructure. The relation between totally different shapes of any onesubstance such as the cubes and octahedra of sodium chloride isless obvious ; but it can be shown that the faces of cubes and octahedraare oriented in precise but different ways with respect to the internalatomic pattern.Two pieces ofinformation about the fundamental atomic pattern maybe deduced from the actual shape of a crystal, provided this crystal


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H IDENTIFICATION OHAP. nshows a sufficient variety effaces and is large enough to permit measure-ments of the angles between the faces. One is a knowledge of the shapeand relative dimensions of the unit of pattern. The other is a partialknowledge of the symmetries of the atomic arrangement.The unit of pattern ('unit cell'). A crystal consists of a largenumber ofrepetitions of a basic pattern ofatoms. Just as in many textilematerials and wall-papers a pattern is repeated over and over again ona surface, so in a crystal a particular grouping of atoms is repeated

c

cf


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CHAP, n SHAPES OF CRYSTALS 15should have arrived at the same shape EFGH for the unit of pattern ;the position ofthe origin does not matter. Note that IJKL may equallyclaim to be the unit of pattern, inasmuch as it contains one unit ofpattern and has exactly the same area as ABCD or EFGH ; and manyother still more elongated areas, each containing one unit of pattern,could be drawn ; in practice, however, it is usually most convenient toaccept as the unit of pattern the area with the shortest sides, that is tosay, the area most nearly* approaching rectangular shape.

All patterns on surfaces can be divided into similar areas in this way,and the unit of pattern is always a parallelogram. The shape and dimen-sions of the parallelogram vary in different ways ; it is possible to havesquare units, rectangular units with ur^equal sides, and non-rectangularunits with either equal or unequal sides.In a crystal we can do the same thing in three dimensions. Again the

choice of origin does not matter, and again we can divide the wholestructure into units (of volume this time) by joining similarly situatedpoints by straight lines. Fig. 5 show$ the arrangement of the ions in acrystal of caesium bromide. Any caesium ion has exactly the samesurroundings as any other, and if the centre of each is joined to thecentres of its nearest neighbours, the whole structure is found to bedivided into cubes, each of which has caesium ions at its corners and abromine ion at its centre. The centre of a bromine ion might equallywell have been selected as the origin, and then the cubic units ofpatternwould have bromine ions at their corners and caesium ions at theircentres. (Note that no bromine ion 'belongs' specifically to any onecaesium ion ; its relations to the eight caesium ions surrounding it areequal. There are thus no 'molecules' ofCsBr in the crystal ; the structureis simply a stack of positively charged caesium ions and negativelycharged bromine ions.)

Fig. 5 also shows an example of a molecular structure that ofhexamethylbenzene, C6(CH3 )6 . The molecules, which can be representedas disks, are all stacked parallel to each other, and if the centre of eachmolecule is joined to those of its nearest neighbours, the structure isdivided into a number of identical units of pattern, each of which is anon-rectangular 'box' with all three sets of edges unequal in length.The unit of pattern in a crystal is always a 'box' bounded by threepairs of parallel sides. The shape and dimensions of the box, that is,the lengths of its three different sorts of edges ('axes') and the anglesbetween them, are characteristic for each different crystal species ; insome crystals the box is a cube, in others it is rectangular with unequal


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planeBCHEplane ADGF

plane DCF6

plane BCF(100)

plane ABCD

planeAFH

plane EBO \\.;/ plane AFC

FIG. 5. a. Caesium bromide, CsBr. Loft, structure. Right, shape of crystal. 6. Hexu-methylbenzene, C,(CH,) 6 . Left, structure. Right, sha.po of crystal, c. Coppor. Loft,

structure. Right, shape ot crystal.


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CHAP. II SHAPES OF CRYSTALS 17edges, in others the angles are not right angles, and so on. We shall notat this point catalogue the various types of shape ; we merely observethat various shapes of pattern-unit are possible ; the crystal structure ofcaesium bromide represents the most highly symmetrical and that ofhexamethylbenzene the least symmetrical of the possible shapes.

It is sometimes more appropriate to use for purposes of reference abox containing more than one unit of pattern. For instance, in crystalsof metallic copper the atoms are arranged in the manner shown inFigs. 5 c and 6. All the atoms have precisely the same surroundings, and

FIG. 6. Face-centred cubic unit coll of roj>|x*r (left), and body -centred culm; unit cell ofa iron (both shown by broken lines). In each case a unit containing one pattern-unit(one atom) is heavily outlined.

the true unit of pattern, formed by joining similarly situated points so asto divide the structure into 'boxes' with atoms at the corners only, is theheavily outlined rhombohedron in Fig. 6 ; there is one atom, one pattern-unit, to each 'box'. (One at each corner of the box makes eight in all ;but each one is shared between the eight boxes which meet at the corner ;therefore each box has the volume of one pattern-unit .) But it is foundthat atoms A,B,C 9 D, E, F, G, and // fall at the corners of a cube, andatoms /, 7, K, L, M, and N in the centres of the faces of the same cube.This cube is accepted as the unit cell, in spite ofthe fact that it containsfour pattern-units comprising one copper atom each. (The corner atomscount as one to each cube ; the six atoms in the face centres are eachshared between two cubes; thus the number of atoms per unit cubeis 1+ 3 = 4.) There are two reasons for this. The first and more im-portant reason is that the symmetries of the complete arrangementare the same as those of crystals in which the shape of the true pat-tern-unit is cubic; crystal symmetry will not be discussed here an

4458


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18 IDENTIFICATION CHAP, nintroductory account of it is given later in this chapter. The secondreason for accepting the four-atom cube as the unit cell is that a cube isa more convenient frame of reference than a rhombohedron. This parti-cular 'compound' unit cell is described as 'face-centred'. Other types of'compound' unit cell are the body-centred, with identical pattern-unitsin the centres of the cells as well as at the corners (see the structure of airon in Fig. 6), and the side-centred, with identical pattern-units at thecentres of one pair of opposite faces in addition to those at the corners.The arrangement of the pattern-units, the assemblage of points each ofwhich represents one pattern-unit, is called the space-lattice. The pointsof the space-lattice the 'lattice points' are thus corners of the trueunit of pattern ; the conventionally accepted unit cell may be simple orcompound; if compound, it may contain two or more space-latticepoints.We now have to consider the faces of crystals and their relation tothe geometry of the precisely patterned assemblage of atoms whichconstitutes the solid material. This subject is best approached bythinking about the manner in which crystals grow. Crystals usuallyhave plane faces, firstly because they do not grow at the same rate in alldirections, and secondly as a result of the specific manner in whichsolid material is deposited.

Crystal growth . Suppose we had the task ofpacking a large numberof atoms or ions or molecules together to form a predetermined arrange-ment. We should find that the most convenient way of building up thestructure is to arrange one layer of building units, then put a secondlayer on top of the first, and so on. But we should have to choose whichlayer to put down first, and there are many different layers which mightbe selected ; there* are very many ways in which a crystal structurecould be divided into layers by planes passing through it. A few possibleways are shown in Fig. 7. In practice we should choose the 'simplest'possible plane, that is to say, a plane which is as layer-like as possible,a plane in which the building-units atoms in sDme crystals, ions ormolecules in others are packed closely together. Thus, to build thecrystal of hexamethylbenzene (Fig. 5 6), it would obviously be moreconvenient to choose planes such asABCD and DCFG, which are paral-lel to the side of the unit cell, rather than a plane such as BDF, whichis inclined to all the edges of the unit cell, as the basis for our buildingoperation.

This is apparently what happens in nature when a crystal grows froma solution or melt. When growing crystals are watched under the micro-


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FIG. 8. Above: layer formation on crystal of cadmium iodide ( X 600). Below, left: layerformation on crystal of sodium chloride (X1400). Below, right: skeletal growths ofammonium chloride ( X 20).


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n SHAPES OF CRYSTALS 19scope, using a high magnification and dark ground illumination,,layerscan often be seen spreading over the faces one after another (Fig. 8,Plate I) ; sometimes it can be seen that relatively thick layers whichspread at a moderate speed are built up from much thinner, much morerapidly spreading, layers ; and it seems likely that the same thing occurs,down to the molecular or ionic scale the building units arrange them-selves layer by layer. (See also Marcelin, 1918 ; Volmer, 1923 ; Kowarski,

Fio. 7. Dividing a crystal into layers. A few of the simpler ways.(Each dot is a lattice point.)

1935.) And this process occurs only on certain planes ; most crystals arebounded by only a few faces, sometimes all of the same type (for in-stance, in cubic crystals), though more frequently of a few differenttypes ; and in structurally simple crystals these types are always denselypacked planes.

In the hexamethylbenzene crystal the most densely packed planesare those parallel to the unit cell edges, and we find that crystals ofhexamethylbenzene grown from a pure solution in benzene are parallele-pipeda with the three pairs of faces parallel to the faces of the unit cell(Lonsdale, 1929). In caesium bromide (Fig. 5) the most densely packedplanes are those such as ACGE which cut two edges of the unit cell atequal angles and are parallel to the third, and caesium bromide crystals(grown from pure aqueous solution) are rhombic dodecahedra which are


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20 IDENTIFICATION CHAP. IIbounded entirely by such planes (Groth, 1906-19). In crystalline copper(Fig. 5 c) the most densely packed planes are those such as BEG whichcut the three edges of the unit cell symmetrically (note that atomsK , J,and N fall on plane BEG) ; copper crystals grow from the vapour asoctahedra, the faces of which are just these most densely packed planes(Groth, 1906-19).For some of the more complex crystals it is not easy to define plane

density of packing of atoms or molecules : a pfane parallel to a crystalface, taken at any level, passes through many atoms, but it cannot passthrough the centres of more than a small proportion of them. Forinstance, the particular plane of the lead chloride crystal illustrated inFig. 39, if it passes through the cerffcres of the atoms at the corners of themarked unit area, does not pass exactly through the centres of any ofthe other atoms, which lie at various distances above or below the planeof the paper. It would be difficult to say which of these should 'count*in the reckoning ofplane density ofpacking ofatoms. (See Niggli, 1920.)But plane density of lattice points is a precisely defined magnitude ;and it is on this that we must focus our attention for it is found thatthe faces of crystals are always densely packed with lattice points. Inother words, if we regard the group of atoms associated with a latticepoint as the building unit, we may say that the faces of crystals areplanes of high reticular density of building units.

It will be evident that, since the faces are parallel to definite planesof lattice points, the interfacial angles are constant in different crystalsof the same substance. Variations in local conditions during growthmay cause some crystals of hexamethylbenzene, for instance, to belonger or thinner than others in the same batch ; and the eight faces of acopper crystal, which in uniform growth conditions would grow to thesame size, may in practice be found to have very different sizes ; butwhatever the variation in the actual dimensions of crystals ofany parti-cular substance, the interfacial angles are constant, provided that thesame type of face is present.

Sparsely packed planes usually do not appear as faces on growingcrystals, but if we deliberately create such surfaces we can study theirgrowth. Fig. 9 illustrates what happens when a cubic crystal of sodiumchlorate (NaClO3 ) is partly dissolved to a rounded shape so as to presentall possible surfaces, and then put into a supersaturated solution. Thediagram is two-dimensional for the sake of simplicity it is a sectionthrough the middle of the crystal. At first, small faces appear on thecorners of the square section ; but it is found that the rate of growth of


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CHAP. II SHAPES OF CRYSTALS 21these small faces the thickness of solid deposited in a unit of time isgreater than that of the cube faces, and as a result of this, the smallfaces ultimately disappear and the final crystal is entirely bounded bythe most slowly growing faces, the ordinary cube faces. (See also Arte-meev, 1910; Spangenberg, 1928.) This experiment brings out the factthat the faces which appear on growing crystals are those with thesmallest rate of thickening. A small rate of thickening, with perhaps agreat rate of spreading, are the growth characteristics one expects of theplanes with highest reticular densityand widest interplanar spacing.When crystals grow rapidly in stirred,strongly supersaturated solutions (&sthey often do under the usual condi-tions of crystallization in thelaboratoryor in industrial plant) there is a plentifulsupply of solute round each growingcrystal; external conditions are fairlyuniform, and the controlling factor isthe architecture of the crystal. Underthese conditions the picture of crystalgrowth given in the previous para-graphs adequately represents whathappens ;f the crystals are bounded byvery few faces the minimum numberof the most slowly growing 'simple'planes necessary to enclose a solid figure. On the other hand, crystalsof many minerals, for instance, have grown very slowly in very slightlysupersaturated solutions in which the supply of solute is very limitedand may vary locally owing to stagnant conditions, convection currents,the proximity of other crystals, and so on. The external conditions thusplay a large part in determining the shape ; faces which, given equalchances, would grow at different rates may actually grow at the samerate, &nd vice versa. These crystals therefore often show a variety offacets which do not appear on crystals grown rapidly. Subsidiary facetsmay also appear if the temperature of a crystallizing solution fluctuates ;partial dissolution rounds off the crystals, and when growth is resumed,small facets appear on the rounded corners, and these may not havetime or opportunity to eliminate themselves by rapid growth as in Fig. 9.

t Except in extreme conditions (very high supersaturation), when skeletal crystals areformed ; and a few substances grow in skeletal form under ordinary conditions. See later.

FIG. 9. A rounded crystal of sodiumchlorate, on being put into super-saturated solution, develops 110 and1 00 faces. The more rapidly growing1 10 faces are subsequently eliminated.


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22 IDENTIFICATION CHAP, nThe production of beautiful, richly faceted crystals by the simplemethod of leaving a dish of solution for several days on a laboratorybench without temperature control is undoubtedly often due to suchtemperature fluctuations. It is still true, however, that all the faces onsuch richly faceted crystals are fairly simple planes, in the sense thatthey have a fairly high reticular density of lattice points. It is also truethat the principal faces are in general simpler than the subsidiary facets.The shape of a crystal may be modified, or even completely changed,by the presence of certain impurities in the solution (see Fig. 3). Thereason is that the impurities are strongly adsorbed only on certain facesof the crystal, thereby retarding the growth of these faces (Gaubert,1906; Bunn, 1933; Royer, 1934).* The impurity may be adsorbed onfaces which normally grow rapidly (that is, planes which are not thesimplest and do not normally appear), and in these circumstances therate ofgrowth of these faces may be so much reduced that they becomethe predominant faces on the crystal. The presence ofmodifying impuri-ties may often be unsuspected ; hence we sometimes find crystals exhi-biting for no apparent reason faces not of the simplest type.Abnormal external conditions may thus be responsible for an ap-parent breakdown in the principle of simplicity of faces. However,

apparent exceptions to the principle cannot always be attributed to ab-normal external conditions. It is not justifiable to regard the principle ofsimplicity as more than a broad generalization ; that is to say, even whenexternal conditions are normal, the faces on crystals, though alwayssimple, are not necessarily the simplest possible. (See also Niggli, 1920.)The rates of growth of crystal faces are of course determined by thedistribution of the forces between the atoms, ions, or molecules, and itis not to be expected that a purely geometrical generalization (as theprinciple of simplicity is) would cover adequately such complexities.In particular it is to be noted that in ionic crystals the distribution ofelectric charges in the various planes plays an important part (Kossel,1927; Stransky, 1928; Brandes and Volmer, 1931).Nevertheless the broad generalization is of the greatest value ; for we

can measure the angles between the faces of a crystal, 'and, assumingthat these faces are simple that is, they are densely packed with latticepoints and are either parallel to the unit ceil faces or are related in somesimple way to the unit cell we can usually deduce the type of unit cell,and very often calculate its relative dimensions and angles.Not all crystals are solid polyhedra. We may approach the subject ofirregularities in crystals by remarking that when a crystal is growing


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CHAP, ii SHAPES OF CRYSTALS 23from a solution, it sometimes happens that growth in the centres of thefaces stops, while growth in the outer regions of the faces (near the edgesand corners) continues, A hollow is thus formed in the centre of eachface. If, as often happens, the hollow is subsequently closed over,mother liquor is included in the crystal. This may be repeated morethan once, and is a common cause of opacity in crystals, and also of thesubsequent caking of crystalline products when stored. (Mother liquordiffuses out, and deposits solute at the points of contact of crystals,cementing them together.)

If such cavities are not closed over, the final crystals have hollowfaces; often there is a step-formation down each hollow. In extremecases growth is maintained only towards the corners of crystals, leadingto skeletal forms, in which successive branching occurs, as in ammoniumchloride, illustrated in Fig. 8, Plate I ; the directions of growth here arethe axial directions of the cubic unit cell. When crystals grow in thinfilms or droplets of liquid, distortion may occur ; a familiar example isice, which forms irregular tree-like patterns when it crystallizes fromliquid on window panes.

Such tendencies may be reduced by growing crystals very slowly, forinstance by extremely slow cooling or evaporation. In fact, when it isdesired to obtain perfect crystals for goniometric or X-ray work, thegolden rule is to grow them as slowly as possible. Excessive nucleusformation in solutions can often be avoided by removing dust particlesin the following way. A solution saturated at, say, 30 C. is made upand allowed to cool without disturbance to room temperature ; it is thensuddenly disturbed, so that a shower of small crystals is formed ; thesecarry down with them any nucleus-forming particles which were in thesolution. The solution is then filtered, warmed slightly to destroy anynow nuclei formed during filtration, and then left undisturbed to eva-porate slowly.

Another method, often useful for organic substances, is to make a solu-tion in one solvent and to cover this with a less dense liquid in whichthe substance is much less soluble ; crystals grow at the interface. Thetwo solvents must be at least partially miscible.

Sparingly soluble salts which are conveniently formed by precipita-tion reactions may sometimes be induced to form good crystals by adiffusion method. Solutions of the reagents are put in two separatebeakers, both completely filled and standing in a larger vessel ; water iscarefuDy poured in to cover both beakers, and the arrangement is thenleft undisturbed (L. M. Clark: private communication).


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24 IDENTIFICATION CHAP. IIThe amount of structural information obtainable by the morpholo-

gical study of skeletal crystals is naturally very limited, especially whenthey are distorted. In order to be able to deduce the shape of the unitcell it is necessary to ha^ve well-formed polyhedral crystals. The facesof such crystals are, as we have already seen, related in some simple wayto the unit cells. We must now define more closely what is meant by thelast phrase 'related in some simple way to the unit cells' and to do

Flu. 10. Various sots of planes in a crystal.

this it is necessary to give some account of the accepted nomenclatureof crystal planes.Nomenclature of crystal planes. Attention has already beendrawn to the many ways ofdividing a crystal into layers by sets ofplanespassing through lattice points (Fig. 7). Each of these sets of parallelplanes is described by three numbers such as 210 or 132, the meaning ofwhich is best shown by a few examples. For simplicity, think first of allin only two dimensions, that is, look at the crystal along one axis saythe c axis as in Fig. 10. In this diagram the points, each of whichrepresents a row of lattice points one behind the other, are seen to lie onsets of straight lines (planes seen edgewise). Every point lies on one ofthese planes. Now along the axial directions count the number ofplanes


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CHAP. II SHAPES OF CRYSTALS 25crossed between one lattice point and the next ; these numbers are theindex numbers. Thus, for the set of planes in the bottom right-handcorner, three planes are crossed in going along a from one lattice pointto the next, and two planes in going along b from one lattice point to thenext ; the first two index numbers are therefore 32. The third indexnumber is 0, because this set ofplanes is parallel to the c axis, and there-fore no planes are crossed in going along c ; this set of planes is thus the320 set. Other sets of planes, with indices 110, 100, 010, and 120 (allparallel to the c axis), are also illustrated in this diagram.

Fia. 11. This set of parallel planes has indices 312.

A set of planes inclined to all three axes is shown in Fig. 11. Along a,three planes are crossed between one lattice point and the next ; along6, one plane is crossed at each lattice point, and along c, two planes perlattice point : the indices are 312.

Alternatively, one could say that these planes cut the a axis at inter-vals of a/3 (a being the repeat distance in this direction), the b axis atintervals of 6/1, and the c axis at intervals of c/2, the indices beingdefined as the reciprocals of these intercepts. This comes to the samething as the definition already given, and corresponds to that found inmost text-books of crystal morphology ; but it is really simpler to thinkofnumbers ofplanes rather than reciprocals ofintercepts ; andmoreover,the present definition links up with the method of indexing X-rayreflections (see Chapter VI).Each type of plane is a possible crystal face, although in actual fact

only a few simple types of plane usually appear as crystal faces. Thenext sketch, Fig. 12, shows an actual crystal (ammonium sulphate)with the indices of its front faces marked. This sketch will also serve toillustrate the conventions about crystal set-up and positive and negative


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IDENTIFICATION CHAP. IIdirections. In order to show as many faces as possible, crystals aredrawn as seen from a viewpoint inclined to all three axes and defined inthe following way. Imagine first of all the crystal with its c axis verticaland its 010 plane seen edgewise ; now shift the eye a little to the rightand upwards. The c axis still appears vertical, the b axis lies left andright but not quite in the plane of the paper, and the a axis points alittle to the left and downwards as it appears to come out above the

Flo. 13. Indices of planes of hexagonalcrystals. ABCDEFA'B'C'D'F/F', hexa-gonal prism; ABCOA'B'C'O', unit cell.FIG. 12. A crystal of ammonium sulphate ACO^ plane whir}]> in conforinjty with(class rnmm). (After Tutton.) indices of crystals of other systems, iscalled 111. For the sake of treating thethree equivalent directions O/4, OG, andOK equally, this plane i sometimes known

as 1121.paper. Usually perspective drawing is not attempted; most crystaldrawings are orthogonal projections. Positive directions are upwardsalong c, to the right along 6, and forwards (above the paper) along a.Intercepts in the negative directions are represented by minus signsabove the index numbers, thus: 120, 111. Naturally it is sometimesnecessary to depart from the conventional viewpoint to illustrateparticular features of crystals more clearly.An extension of this system of nomenclature is sometimes encoun-tered in descriptions of crystals of hexagonal type (Fig. 13). The unitcell of these crystals has a diamond-shaped base, the a and b axes beingequal in length and inclined to each other at an angle of 120. The caxis is perpendicular to the other two. Although only two horizontal


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CHAP, ii SHAPES OF CRYSTALS 27axes are strictly necessary for purposes of description, nevertheless thereare three horizontal directions, all exactly equivalent, at 120 to eachother ; any two of them could be taken as the a and b axes. In order tobring out this feature, index numbers referring to all three horizontalaxes, as well as the vertical (c) axis, are given, thus: 1121. The lastnumber refers to the c axis, the first three to the horizontal axes. Thethird index, which is always necessarily numerically equal to the sumof the first two but of opposite sign, is really redundant. This nomen-clature will be found in descriptions of the shapes of hexagonal crystals,but for internal crystal planes it is customary to omit the third index.The indices of single crystal faces are sometimes enclosed in brackets,thus: (100); this distinguishes a face from the corresponding set ofinternal planes 100. Curly brackets signify a set of equivalent faces : fora cubic crystal {100} would mean the set 100, TOO, 010, OK), 001, and OOT.The law of rational indices. We have seen that the faces ofstructurally simple crystals, the planes on which deposition of solidoccurs layer by layer, are in general those planes which have a highreticular density of lattice points in each plane and wide interplanarspacing. Sometimes the faces are the planes with the densest packingand the widest interplanar spacing, but there are many exceptions tothis, for various reasons which have already been mentioned. It re-mains true, however, that in all cases the actual faces of a crystal areplanes of high (though not necessarily the highest) reticular density.We may call these the 'simple' planes.

It is evident from Figs. 10 and 1 1 that these planes have small indices ;we may therefore state that the actual faces on crystals are planes withsmall indices. In this form, the generalization is what is known as the'law of rational indices', which says simply that all the faces on a crystalmay be described, with reference to the three axes, by three small wholenumbers. It is frequently found that all the faces of even richly facetedcrystals can be described by index numbers not greater than 3 ; numbersgreater than 5 are very rare.

It is the recognition of the law of rational indices which makes itpossible to deduca*probable unit cell shapes from crystal shapes. (It is,of course, not possible to discover the absolute dimensions ; X-ray orelectron diffraction photographs are necessary for this purpose (ChapterVI).) The general principle is to find that unit cell (its angles and relativedimensions) which will enable us to describe all the faces of the crystalby the smallest whole numbers, and, in particular, the largest faces bythe smallest numbers. There is a further condition: all faces which


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28 IDENTIFICATION CHAP. IIappear to be equivalent (for instance, all the eight faces of a regularoctahedron) are given similar indices, that is, are assumed to be relatedin the same way to the most appropriate unit cell ; in other words, thedirections of unit cell edges are chosen in conformity with'the symmetryof the crystal. We shall return to this subject later in this chapter.Meanwhile, the first step in the attempt ta deduce the angles andrelative dimensions of the unit cell of a crystal from its actual shapeis the accurate measurement of the angles between all the faces of thecrystal.

LIGHT

FIG. 14. Principle of the reflecting goniometer. The adjusting head comprises twoiriutually perpendicular arc movements and two cross movements.

Measurement of interfacial angles, and graphical representa-tion. The most accurate method of measuring the angles betweencrystal faces is an optical one, which makes use of the reflection of lightby the plane faces. The crystal is mounted on the stem of a goniometerhead (Fig. 14) by means of wax, shellac, or plasticine ; a beam of parallellight from the collimator strikes the crystal, which is rotated untilone of its faces reflects the beam into the telescope, which is at anyconvenient angle to the collimator. A suitable sharply defined apertureis provided in the collimator, so that its image can be adjusted accu-rately to the cross-wires of the telescope. The crystal is then rotateduntil the light is reflected by the next face ; the angle through which thegoniometer head has been turned is the angle between the normals ofthetwo faces. It is evident that, in order to get reflections from both facesinto the telescope, the crystal must be adjusted very carefully by means


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CHAP, n SHAPES OF CRYSTALS 29of the arc movements of the goniometer head. This is simplest when thecrystal is mounted on the goniometer head so that one of the face-normals is approximately parallel to one of the arc movements ; this arcis adjusted until the reflection from this face appears accurately on thecross-wires. The crystal is now rotated so that the reflection fromanother face (preferably one which is roughly at right angles to the first)enters the telescope ; by a movement of the second arc this reflection isbrought to the cross-wires.

It is found that, when the reflections from two faces are registeredaccurately on the cross-wires of the telescope, other faces automaticallygive their reflections when the crystal is rotated further ; for instance,all the vertical faces of the ammonium sulphate crystal in Fig. 12 givetheir reflections one after the other as the crystal is rotated round thec axis. Such a set of faces is called a 'zone', and the axis of rotationparallel to all the faces is called the 'zone axis'. All the faces of anycrystal fall on one or other of a few zones, and therefore in order tomeasure all the interfacial angles each of these zone axes in turn mustbe set parallel to the axis of rotation of the goniometer head. On asingle-circle goniometer this must be done by remounting the crystalfor each zone ; but two-circle goniometers which obviate the necessityof such re-setting are also obtainable.

It is often useful to be able to represent precisely on a flat surface thethree-dimensional relations between the interfacial angles. The mostconvenient projection for most purposes is the stereographic projection,which is derived in the following way. From a point within the crystalimagine lines drawn outwards normal to all the faces (Fig. 15). Roundthe crystal describe a sphere having the point as its centre. The positionsat which the face-normals meet the surface of the sphere are known asthe poles of the faces. The crystal is thus replaced by a set of points onthe surface of the sphere, each point representing the orientation of acrystal face. In this way we have left behind the actual shape of thecrystal, with the irregularities arising from the conditions of growth,and are now dealing simply with the orientations of faces that is, withthe orientations oflattice planes, which are related in a simple way to theunijj cell. The sphere is now projected on to a selected plane the equa-torial plane in Fig. 15 b by joining all points on its upper half to the'south pole' and all points on its lower half to the 'north pole'. The greatadvantage of this projection (Fig. 15c) is that all zones of faces falleither on arcs of circles or else on straight lines, a circumstance whichmuch facilitates graphical construction. (Each such arc or straight line


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30 IDENTIFICATION CHAP. II

FIG. 15. The stereographic projection.

passes through opposite pointson the equatorial circle.) Polesin the northern hemisphere aredenoted by dots, those in thesouthern hemiphere by littlerings.For further information on

stereographic projections andthe spherical trigonometrynecessary for handling gonio-metric data, books by Miers(1929), Tutton (1922), andBarker (1922) may be consulted.Deduction of possible unitcell shape from crystalshape. Preliminary. In thisbook we are concerned chieflywith optical and X-ray methods,and we shall consider crystalmorphology only so far as isnecessary for the full use of suchmethods for identification or forstructure determination. Butalthough it is not intendedto deal with morphologicalmethods in a quantitative way,it is very necessary to considerin rather more detail the rela-tion between the external shapeof a crystal and that of its unitcell ; and this subject is perhapsbest developed in the guise ofa consideration of the problemof deducing the probable unitcell shape from the externalshape of a crystal. We have al-ready seen that the principle onwhich the attempt is based isthe principle of simplicity ofindices, coupled with the con-


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CHAP, ii SHAPES OF CRYSTALS 31formity of the indices with the symmetry of the crystal. We now seehow this principle can be applied in practice. First of all, we shall seethe principle of simplicity in action by itself; and we shall then findit necessary to consider crystal symmetry in some detail.The planes with the simplest indices 100, 010, and 001 are thosewhich are parallel to the sides of the unit cell, and we find that on manycrystals these form the principal faces, and on some crystals (especiallythose grown rapidly m,stifongly supersaturated solutions) the only faces.One example, hexamethylbenzene, has already been given; it formsnon-rectangular parallelepipeda with the three pairs of faces parallelto the unit cell faces. Another example is anhydrite, CaS04 ; the unitcell of this crystal is a rectangular box*with unequal edges, and it growsas a rectangular brick with unequal edges, though it must be emphasizedthat the relative dimensions of the crystal itself have no direct con-nexion with the dimensions of the unit cell. (The rates of growth of thevarious faces of any crystal depend, in the first place, on the forcesbetween the atoms, ions, or molecules in different directions, and theseforces have no direct connexion with the unit cell dimensions ; moreover,these rates of growth are affected by external conditions.) Such crystalstell us the angles of the unit cell, but they do not tell us anything aboutthe relative dimensions of the unit cell edges.

If we are to be able to calculate the relative dimensions of the unitcell of any crystalline substance, some of the faces on the crystals mustbe inclined to the faces of the unit cell. Suppose we have a crystal of theshape shown in Fig. 16 a a rectangular brick with the (unequal) edgesbevelled (an orthorhombic crystal). We naturally assume that the faceswhich are perpendicular to each other are parallel to the faces of the unitcell, which is evidently a rectangular box. The indices of the principalfaces are thus assumed provisionally to be 100, 010, and 001. Thesimplest indices for the faces which bevel the edges are 110, Oil, and101. If we assume that a face is Oil, we are assuming that successiveidentical planes of lattice points parallel to this face are parallel to thea axis, and that in passing along either b or c, only one plane is crossedin the interval between one lattice point and the next. (See Fig. 16 6.)It is evident that c/b = cot 0. In the same way, by assuming thatanother face is 110, we can obtain a/6 ; and this settles the shape of theunit cell and the indices of the remaining faces ; thus, the third differentbevelling face might turn out to be, not 101 as first suggested, but 201or 102. If our crystals also have faces cutting off the corners (Fig.16 c), the indices of these faces can be found (by slightly more complex


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IDENTIFICATION CHAP. IItrigonometry) from the angles between these 'corner' faces and theprincipal faces.

Alternatively, it might have been assumed initially that these 'corner*faces are 111, III, and so on ; this assumption would have given us a setof axial ratios, from which the indices of the bevelling faces could bededuced.

It is always possible to find alternative sets of indices, correspondingto different axial ratios, for any crystal. Thus1,' consider the ammonium

(b)c-UNIT

: CELL010

FIG. 16. Determination of the probable shape of the unitcell from interfacial angles.

sulphate crystal (Fig. 12), which, like the example just given, has arectangular unit cell. Let us call the faces 110, Oil, 130, 021, and 111P> ?> P'> q'> and o respectively. If it had been assumed that q' is 01 1 andp 110, then this group effaces would be 110, 012, 130, Oil, and 112.Or it might have been assumed that p 1 is 110 and q Oil, in which casethe group of faces would be, 310, Oil, 110, 021, 311. But the sets ofindices given by the second and third schemes are less simple than thoseresulting from the first assumptions, and therefore the axial ratiosderived in the first scheme are accepted as the probable relative dimen-sions of the unit cell edges. This turns out to be correct.Here we have the key to morphological crystallography. The principlefollowed throughout is to find that unit cell shape which, subject to thecondition that similar faces shall have similar indices, will allow all the


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CHAP, ii SHAPES OF CRYSTALS 33faces of a crystal to be indexed by the smallest possible whole numbers,the principal faces being given, as a general rule, the simplest indices.This method was developed during the last century, long before X-rayswere discovered, though the term 'unit cell' was not used. The set ofaxes deduced in this way was regarded primarily as the most convenientframe of reference for the accurate description and classification of anycrystal. Nevertheless it is clearly more than a convenient frame ofreference ; it corresponds'to some fundamental feature of the ultimatestructure of the crystal. We know now, as the result of the study of theatomic structure of crystals by X-ray methods, that the relative axialdimensions deduced by morphological methods are in fact very oftenthe exact relative dimensions of the u&it cell. Even when they are notcorrect, there is always a very simple relation between the 'morpholo-gical' unit and the true unit ; one of the morphological axes is perhapstwice as long or halfas long (in relation to the other axes) as it should be.This obviously means that the principle ofsimplest indices is not strictlytrue for these crystals ; some of the faces on these crystals are, so tospeak, not the simplest but the next in order of simplicity. There is nodoubt about the general soundness of the principle of simplest indices,but it is not a rigid law.The examples given hitherto have been particularly simple ones,because some of the faces have been at right angles to each other, andthis has given the clue to the type of unit cell. But many crystals donot possess faces parallel to the unit cell faces, and for such crystalsthe type of unit cell, and possible indices for the principal faces, arevery often not by any means obvious. To approach such problems it isnecessary to introduce the all-important subject of crystal symmetry.The type of unit cell is entirely bound up with the symmetry of theatomic arrangement ; it is, in fact, the symmetry of the atomic arrange-ment which decides which (if any) of the unit cell angles shall be rightangles, and how many of its edges shall be equal. Therefore if we canrecognize the symmetries ofany particular crystal, this leads us at onceto the unit cell type and to the probable directions of unit cell edges.And this is not all. Each type of unit cell may arise from a number ofdifferent types of atomic arrangement, and some of the symmetrycharacteristics of these different types of atomic arrangements are re-vealed by shape-symmetries. In classifying crystals we can first of alldivide them into several systems according to unit cell types, and theneach system can be divided into several classes according to those sym-metry characteristics which are revealed by shape. The consideration

4458


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34 IDENTIFICATION CHAP, nof crystal symmetry may thus take us further than the mere derivationof unit cell type.

Internal symmetry and crystal shape. Consider first one of thesimplest and most highly symmetrical of atomic arrangements, thatwhich is found in crystals of sodium chloride and in many other simplebinary compounds. The atomic arrangement is shown in Fig. 17 a. Theunit cell is a cube ; if we take the corner of the unit cell to be the centreof a sodium ion, there are also sodium ions at the centre of each face,the lattice being a face-centred one; the chlorine ions are half-wayalong the edges and also in the centre of the unit cell. Note first thattlie reason why the three mutually perpendicular axes are equal inlength is that the arrangement of atoms is precisely the same along oneaxis as it is along the other two ; the 100 plane has exactly the samearrangement of atoms as the 010 and 001 planes ; secondly, that whena sodium chloride crystal grows in a pure solution, it is inevitable that,provided the three types of faces have the same chance (in a stirredsolution, for instance), they grow at the same rate, and the crystalbecomes a perfect cube.

If sodium chloride crystals are grown in a solution containing 10 percent, of urea, they grow as regular octahedra ; but although the externalshape is different from that of crystals grown from a pure solution, theinternal structure is exactly the same; the same internal lattice isbounded by surfaces of a different type in the two sorts of crystals.The octahedral faces (111, 111, lTl,Tll, iTT, TlT, TTl, and TTT) are per-pendicular to the cube diagonals ; the atomic arrangement on all octa-hedral faces is the same, and if we proceed from any point in the crystalalong any of the eight diagonal directions, we shall come across thesame atomic distribution (alternate layers of sodium and chlorineions) ; consequently, in uniform growth conditions all the octahedralfaces grow at the same rate, and the crystals grow as perfectly regularoctahedra.Now although the cube and the regular octahedron are quite differentsolid shapes, yet their symmetries are exactly the same ; and it can beseen (in Figs. 17-20) that the symmetries of these solid figures are thoseof the arrangement ofatoms in a, sodium chloride crystal. Rotate a cubeabout an axis perpendicular to one of its faces and passing through itscentre (Fig. 176); after a quarter of a turn it presents exactly the sameappearance as it did at first ; after half a turn, again the same appear-ance, and likewise after three-quarters of a turn ; in fact, it presents thesame appearance four times during a complete revolution ; the axis is an


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CHAP. II SHAPES OF CRYSTALS 35axis of fourfold symmetry. There are three such fourfold axes, all atright angles to each other and parallel to the cube edges. A regularoctahedron likewise has three fourfold axes, passing through its corners(Fig. 17 c). These fourfold axes correspond with those of the atomic

(a)

FIG. 17. a. The atomic arrangement in sodium chloride, and some of its axesof symmetry, b and c. Fourfold axes of cube and octahedron, d and e. Twofoldaxe of cube and octahedron.

arrangement ; every line which passes through a row of atoms parallelto a unit cell edge is an axis of fourfold symmetry, since the atomicarrangement (regarded as extending indefinitely in space) presents thesame appearance four times during a complete revolution round thisline. Similarly there arc, passing through the edges of both cube andoctahedron, six axes of twofold symmetry, involving identity of appear-ance twice during a complete revolution (Fig. 17 d and e) ; and finally,passing through the cube corners and perpendicular to the octahedron

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